Some common accelerometer/gyro chips have built-in processor that performs integration for you. An example I am familiar with is MPU-6050.
Given that capabilty, you can get the ready-made data for current direction of your chip directly from the chip using I2C.
Please search for "mpu 6050 dmp" to see some examples.
Expanding on the answer given by C.O Park and comment by Ben.
The gyroscope and accelerometer will give you 6 sensor values corresponding to the $x$/$y$/$z$ linear accelerations and the $x$/$y$/$z$ angular velocities. If you are looking to interpret whether your robot is turning left or right at any particular moment then you can simply check the sign of the ...
Just monitor the rotation velocity of z-axis (z value from gyroscope or any other axis according to your setup).
If the value is over some threshold and the sign is positive, it is turning left. If the sign is negative it is the right turn.
The right way to do this is to accumulate the value stream and see if the accumulated angle is near to 90 or -90 degree....